Consider the diffusion equation with convection on the real line. მu. at = K J²u მ2 მu. + C- მე u(0,x) = f(x) (∞0>x>∞0-) a) Take the Fourier Transform of the PDE and initial data with respect tox and solve for u(t, w). b) Using the fact that the inverse Fourier Transform of a Gaussian is F(w) = e-aw² f(x) = -x²/(4α) (a > 0) write the solution u(t, x) to the convection equation as a convolution of the initial data f(x) with an appropriate function. (HINT: You should get a term like eicut in u. This should simply shift x to x + ct in the final answer!)

Show that t, e^t, and sin(t) are linearly independent.

Express dw/ðu and əw/dv as functions of u and v both by using the Chain Rule and expressing w directly in terms of u and v before differentiating. w = In(x? + y? + z²), x= e" sinu, y=e" cos u, z= ue ": (u,v) = (-2,0)

EX: Find the F.T. of the triangular pulse: g(t) = AA (+) Sols Apply differentiation property of F-T g(t) = 2A S(t+1)_4A S(t) + 2A S(t-I) The F. I for the both sides gives. (jw)² G (w) = 2A (NE_2 +2²) (e j들 (JW)³&(W) = 4A (2 ² + 2² (e 2 G(W) = -4A whe - 8A w²t -₁) 8A ( cos 뜰 -1) = 82 (1-005) 2 sin wt 4 At sin wt 2 Wt 4 2 3/4 N 2A T ZA 0 0 8(1)-A(+) =9c How did he get the value inside the circle Hes -9C2A ہم تم at ot

Find the velocity and acceleration vectors and the equation of the tangent line for the curve r(t) = √3ti + e¹¹j + 6e¯¹k at t = 0. (Use symbolic notation and fractions where needed. Give your answers in the form (*,*,*).) v(0) = a(0) = (Use symbolic notation and fractions where needed. Give your answers in the form (*,*,*). Use t for the parameter that takes all real values.) 1(t)

Find 4 dx J-cos(3пx) (ез + 1)dtr.

Find the derivative of f(w)= sin(4w²)+cos(4w²). A) Swcos(4w°)–8wsin(4w²) B) -8wcos(4w?)+8wsin(4w²) C) 4cos(4w?)–4sin(4w?) D) 4cos(4w?)+4sin(4w²)

Find the velocity and acceleration vectors and the equation of the tangent line for the curve r(t) = 3 cos(t)i + sin(6t)j at t 0. (Use symbolic notation and fractions where needed. Give your answer in the form (*,*).) v(0) 1(t) = Incorrect (0,6) (Use symbolic notation and fractions where needed. Give your answer in the form (*,*). Use t for the parameter that takes all real values.) = a(0) = (3,0) +t(0,6) (-3,0)

Suppose the velocity of a ninja is given by v = ft, 3t², sin(π) at time t. If the position of the ninja is (1, 0, 0) at time t = 1 then find the position and acceleration of the ninja as a function of time t.

Use the chain rule